Lemma: Smooth parameter dependence

For fixed $\epsilon$, ${{\Lambda }_A}^{\mu }(z,y)$ is jointly smooth on a neighborhood of $M_T\times M$. The polarized second kernel

is jointly smooth as well. Both vanish in each input variable outside $\mathrm{supp}g$.

Proof. The source $H_{A,y}(u)$ is jointly smooth and has a common compact $u$-support when $y\in \mathrm{supp}g$. Parameter differentiation commutes with the wave problem by uniqueness and the differentiated energy estimates of Lemma lem:Cauchy. Thus (eq:Lambda) is jointly smooth.

Every source in (eq:Vbil) is a finite sum of products and derivatives of such smooth parameter families. The same argument applies to $\Omega$. The factor $g(y)$, respectively $g(y)g(y^{\prime })$, proves the input support assertion.